Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

2012-09-10T16:22:43Z

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((\mathbb{C}[\mathfrak{g^*}]\otimes \Omega^{\bullet}{M})^G, d_G)$ (for the definition of equivariant cohomology we can look at chapter 1 and 4 of Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory"). $K< G$ is a closed subgroup, Let $M^K$ be the points of $M$ which has isotropy groups conjugated to $K$, obviously $M^K$ is a $G$-submanifold of $M$ and let $~i: M^K \rightarrow M$ denote the inclusion map. we have a version of localization theorem, see Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory" chapter 11, especially Theorem 11.4.3 in page 178. In more details :

Consider the equivariant cohomology $ H_G(M)$ and $H_G(M^K)$ as $ S( \mathfrak{g^* })^G $ modules. Then the pullback map $$ i^*: H^ * _G(M)\rightarrow H^ *_G(M^K) $$ is an isomorphism after localizing at some certain prime ideals of $ S( \mathfrak{g^* })^G $.

On the other hand, we have the equivariant K-theory $K_G(M)$ and we also have the localization theorem in this side, see Segal "Equivariant K-theory" (1967) section 4, proposition 4.1, which also claims that Then the pullback map $$ i^*: K^ * _G(M)\rightarrow K^ *_G(M^K) $$ is an isomorphism after localizing at some certain prime ideals of $R(G)$, the representation ring of $G$.

We notice the similarity of the above two version of localization theorems. Nevertheless equivariant cohomology and equivariant K-theory are different. The first is the cohomology of a differential graded algebra and the latter is the Grothedieck group of modules of the cross product algebra $G \ltimes C(M)$.

My question is: is there any deep relation between them? Are they valid because of the same reason?

Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory - MathOverflow

Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory